Unit #10 : Graphs of Antiderivatives, Substitution Integrals Goalsmath121/Notes/notes10.pdfUnit #10...

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Unit #10 : Graphs of Antiderivatives, Substitution Integrals Goals: Relationship between the graph of f (x) and its anti-derivative F (x) The guess-and-check method for anti-differentiation. The substitution method for anti-differentiation.

Transcript of Unit #10 : Graphs of Antiderivatives, Substitution Integrals Goalsmath121/Notes/notes10.pdfUnit #10...

Page 1: Unit #10 : Graphs of Antiderivatives, Substitution Integrals Goalsmath121/Notes/notes10.pdfUnit #10 : Graphs of Antiderivatives, Substitution Integrals Goals: Relationship between

Unit #10 : Graphs of Antiderivatives, Substitution Integrals

Goals:

• Relationship between the graph of f (x) and its anti-derivative F (x)

• The guess-and-check method for anti-differentiation.

• The substitution method for anti-differentiation.

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Graphs of Antiderivatives - 1

The Relation between the Integral and the Derivative Graphs

The Fundamental Theorem of Calculus states that∫ b

a

f (x) dx = F (b)− F (a)

if F (x) is an anti-derivative of f (x).Recognizing that finding anti-derivatives would be a central part of evaluatingintegrals, we introduced the notation∫

f (x) dx = F (x) ⇔ d

dxF (x) = f (x)

In cases where we might not be able to find an anti-derivative by hand, we can atleast sketch what the anti-derivative would look like; there are very clear relation-ships between the graph of f (x) and its anti-derivative F (x).

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Graphs of Antiderivatives - 2

Example: Consider the graph of f (x) shown below. Sketch two possibleanti-derivatives of f (x).

0

2

4

6

−2

−4

1 2 3 4

x

f(x)

0

4

−4

1 2 3 4

x

F (x)

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Graphs of Antiderivatives - 3

Example: Consider the graph of g(x) = sin(x) shown below. Sketch twopossible anti-derivatives of g(x).

x

g(x)

x

G(x)

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Graphs of Antiderivatives - 4

The Fundamental Theorem of Calculus lets us add additional detail to the anti-derivative graph: ∫ b

a

f (x) dx = F (b)− F (a) = ∆F

What does this statement tell us about the graph of F (x) and f (x)?

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Graphs of Antiderivatives - 5

Re-sketch the earlier anti-derivative graph of sin(x), find the area underneathone “arch” of the sine graph.

x

g(x)

x

G(x)

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Antiderivative Graphs - Example 1 - 1

Example: Use the area interpretation of ∆F or F (b)− F (a) to constructa detailed sketch of the anti-derivative of f (x) for which F (0) = 2.

1

2

3

4

−1

−2

−3

−4

1 2 3 4−1−2−3−4

x

f(x)

0 1 2 3 4−1−2−3−4

x

F (x)

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Antiderivative Graphs - Example 2 - 1

Example: f (x) is a continuous function, and f (0) = 1. The graph of f ′(x)is shown below.

0

1

2

−1

−2

1 2 3 4 5 6

x

f ′(x)

Based only on the information in the f ′ graph, identify the location and typesof all the critical points of f (x) on the interval x ∈ [0, 6].

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Antiderivative Graphs - Example 2 - 2

Sketch the graph of f (x) on the interval [0, 6]. Reminder: f (0) = 1.

0

1

2

−1

−2

1 2 3 4 5 6

x

f ′(x)

0

1

2

−1

−2

1 2 3 4 5 6

x

f(x)

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Antiderivative Graphs - Example 3 - 1

0 1 2 3 4 5

−2

−1

1

2

Question: Given the graph of f (x) above, which of the graphs below is ananti-derivative of f (x)?

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

(a) (b) (c) (d)

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Integration Method - Guess and Check - 1

We now return to the challenge of finding a formula for an anti-derivative function.We saw simple cases last week, and now we will extend our methods to handle morecomplex integrals.

Anti-differentiation by Inspection: The Guess-and-Check Method

Often, even if we do not see an anti-derivative immediately, we can make an edu-cated guess and eventually arrive at the correct answer.

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Integration Method - Guess and Check - 2

Example: Based on your knowledge of derivatives, what should the anti-

derivative of cos(3x),

∫cos(3x) dx, look most like?

(a) cos(x) + C

(b) sin(x) + C

(c) cos(3x) + C

(d) sin(3x) + C

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Integration Method - Guess and Check - 3

Evaluate

∫cos(3x) dx by guessing and checking your antiderivative.

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Integration Method - Guess and Check - 4

Example: Find

∫e7x−2 dx.

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Integration Method - Guess and Check - 5

Example: Both of our previous examples had linear ‘inside’ functions. Hereis an integral with a quadratic ‘inside’ function:∫

xe−x2dx

Evaluate the integral.

Why was it important that there be a factor x in front of e−x2

in this integral?

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Integration Method - Substitution - 1

Integration by Substitution

We can formalize the guess-and-check method by defining an intermediate variablethe represents the “inside” function.

Example: Show that

∫x3√x4 + 5 dx =

1

6(x4 + 5)3/2 + C.

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Integration Method - Substitution - 2 ∫x3√x4 + 5 dx =

1

6(x4 + 5)3/2 + C

Relate this result to the chain rule.

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Integration Method - Substitution - 3

Now use the method of substitution to evaluate

∫x3√x4 + 5 dx

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Substitution Integrals - Example 1 - 1

Steps in the Method Of Substitution

1. Select a simple function w(x) that appears in the integral.

• Typically, you will also see w′ as a factor in the integrand as well.

2. Finddw

dxby differentiating. Write it in the form . . . dw = dx

3. Rewrite the integral using only w and dw (no x nor dx).

• If you can now evaluate the integral, the substitution was effective.

• If you cannot remove all the x’s, or the integral became harder instead of eas-ier, then either try a different substitution, or a different integration method.

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Substitution Integrals - Example 1 - 2

Example: Find

∫tan(x) dx.

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Substitution Integrals - Example 1 - 3

Though it is not required unless specifically requested, it can be reassuring to checkthe answer.Verify that the anti-derivative you found is correct.

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Substitution Integrals - Example 2 - 1

Example: Find

∫x3ex

4−3dx.

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Substitution Integrals - Example 3 - 1

Example: For the integral, ∫ex − e−x

(ex + e−x)2dx

both w = ex− e−x and w = ex + e−x are seemingly reasonable substitutions.Question: Which substitution will change the integral into the simpler form?

(a) w = ex − e−x

(b) w = ex + e−x

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Substitution Integrals - Example 3 - 2

Compare both substitutions in practice.∫ex − e−x

(ex + e−x)2dx

with w = ex − e−x with w = ex + e−x

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Substitution Integrals - Example 4 - 1

Example: What substitution is most likely to be helpful for evaluating theintegral ∫

sin(x)

1 + cos2(x)dx ?

(a) w = sin(x)

(b) w = cos(x)

(c) w = 1 + cos2(x)

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Substitution Integrals - Example 4 - 2

Evaluate

∫sin(x)

1 + cos2(x)dx.

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Substitutions and Definite Integrals - 1

Using the Method of Substitution for Definite Integrals

If we are asked to evaluate a definite integral such as∫ π/2

0

sinx

1 + cos2 xdx ,

where a substitution will ease the integration, we have two methods for handlingthe limits of integration (x = 0 and x = π/2).

a) When we make our substitution, convert both the variables x and the limits(in x) to the new variable; or

b) do the integration while keeping the limits explicitly in terms of x, writing thefinal integral back in terms of the original x variable as well, and then evaluating.

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Substitutions and Definite Integrals - 2

Example: Use method a), converting the bounds to the new variable, toevaluate the integral ∫ π/2

0

sinx

1 + cos2 xdx

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Substitutions and Definite Integrals - 3

Example: Use method b) method, keeping the bounds in terms of x, andconverting back to x’s, to evaluate∫ 64

9

√1 +√x√

xdx .

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Simplifying Substitutions Integrals - 1

Simplifying Substitution Integrals

Sometimes trying a substitution will still simplify the integral, even if you don’tsee an obvious cue of “function and its derivative” in the integrand.Example: Find ∫

1√x + 1

dx .

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Simplifying Substitutions Integrals - 2 ∫1√x + 1

dx